How do you find the fraction equivalent of a repeating decimals that also includes
digits that do not repeat? Doctor Greenie tweaks a student's formal algebraic
approach, introduces another wrinkle grounded in the base ten number system, then
shares two algorithms based on his work with high school students who take part in
math competitions.

The sum of 1/2, 1/3, and 1/4 of the enrollment of the Business School
is exactly the enrollment of the Language School. The sum of 1/5, 1/6,
1/7, and 1/8 of the enrollment of the Business School is exactly that
of the Design School. What are the enrollments of these three schools,
assuming no school has more than 1000 pupils?